ofex.hamiltonian

This module provides implementations of various quantum physics and chemistry models.

The module includes: - Functions for defining 1D Pauli Hamiltonian models (zz_1d, heisenberg_1d, and heisenberg_1d_ring). - A class PolyacenePPP for representing and working with polyacene molecules using the Pariser-Parr-Pople (PPP) model.

ofex.hamiltonian.zz_1d(num_qubits: int) → QubitOperator

Generate the Z-Z interaction Hamiltonian for a 1D chain of qubits.

Parameters:

num_qubits (int) – The number of qubits in the system.

Returns:

The QubitOperator representing the Z-Z Hamiltonian.

Return type:

of.QubitOperator

ofex.hamiltonian.heisenberg_1d(num_qubits) → QubitOperator

Generate the Heisenberg Hamiltonian for a 1D open chain of spins.

Parameters:

num_qubits (int) – The number of qubits in the system.

Returns:

The QubitOperator representing the Heisenberg Hamiltonian.

Return type:

of.QubitOperator

ofex.hamiltonian.heisenberg_1d_ring(num_qubits) → QubitOperator

Generate the Heisenberg Hamiltonian for a 1D closed ring of spins.

Parameters:

num_qubits (int) – The number of qubits in the system.

Returns:

The QubitOperator representing the Heisenberg Hamiltonian on a closed ring.

Return type:

of.QubitOperator

class ofex.hamiltonian.PolyacenePPP(n_ring: int, param_type: str = 'standard', t0: float = 0.08819833952795732, t_: float = 0.08819833952795732, bond_length: float = 1.4)

The PolyacenePPP class represents a linear polyacene molecule described using the Pariser-Parr-Pople (PPP) model for quantum chemistry calculations. It provides methods for computing molecular Hamiltonians, electron-electron repulsion, hopping terms, and other properties based on the PPP approximation.

Reference:

Chakraborty, H., & Shukla, A. (2013). Pariser–Parr–Pople Model Based Investigation of Ground and Low-Lying Excited States of Long Acenes. The Journal of Physical Chemistry A, 117(51), 14220–14229. doi:10.1021/jp408535u

n_ring

The number of aromatic rings in the polyacene molecule.

Type:

int

t0

Hopping strength value for horizontal bonds.

Type:

float

t_

Hopping strength value for vertical bonds.

Type:

float

bond_length

The bond length between adjacent carbon atoms (in Angstroms).

Type:

float

atom_dist(i: str, j: str) → float

Calculates the Euclidean distance between two atoms based on their coordinates.

Parameters:

• i (str) – The label of the first atom (e.g., “U0” or “L1”).

• j (str) – The label of the second atom (e.g., “U1” or “L0”).

Returns:

The Euclidean distance between the two atoms in Angstroms.

Return type:

float

property atom_name: List[str]

Returns: List[str]: A list of atom names in the polyacene molecule.

ee_repulsion() → FermionOperator

Constructs the electron-electron repulsion operator, including on-site and Ohno interaction terms.

Returns:

The total electron-electron repulsion operator.

Return type:

FermionOperator

fermion_hamiltonian() → FermionOperator

Constructs the full fermionic Hamiltonian of the molecule, which includes contributions from horizontal hopping, vertical bridge hopping, and electron-electron repulsion.

Returns:

The full fermionic Hamiltonian of the molecule.

Return type:

FermionOperator

fermion_symmetries() → List[List[Tuple[int, int]]]

Generates symmetry transformations for the fermionic orbitals in the system based on spatial symmetry operations.

Returns:

A list of symmetry transformations,

where each transformation is represented as a list of tuples. Each tuple defines a permutation of two orbitals (given by their indices). The transformations included are: - σ(x): Reflection across the x-axis. - σ(y): Reflection across the y-axis.

Return type:

List[List[Tuple[int, int]]]

get_molecular_hamiltonian()

Retrieves the molecular Hamiltonian for the polyacene molecule.

Returns:

The fermionic Hamiltonian of the molecule.

Return type:

FermionOperator

hf_state(parity: bool = True)

Constructs a Hartree-Fock (HF) state for the molecule based on a given parity.

Parameters:

parity (bool) – Determines whether the alternating occupation starts with the 0th orbital or the 1st orbital. If True, the occupation starts from the 1st orbital; if False, the occupation starts from the 0th orbital.

Returns:

The Hartree-Fock state as a dictionary where the key is the

BinaryFockVector representing the state and the value is its coefficient.

Return type:

Dict[BinaryFockVector, float]

hopping_bridge() → FermionOperator

Constructs the hopping operator for vertical bonds between the upper and lower parts of the molecule.

Returns:

The bridge hopping operator scaled by -self.t_.

Return type:

FermionOperator

hopping_lower() → FermionOperator

Constructs the hopping operator for horizontal bonds in the lower part of the molecule.

Returns:

The lower horizontal hopping operator scaled by -t0.

Return type:

FermionOperator

hopping_upper() → FermionOperator

Constructs the hopping operator for horizontal bonds in the upper part of the molecule.

Returns:

The upper horizontal hopping operator scaled by -t0.

Return type:

FermionOperator

inv_spatial_idx(i: int) → str

Converts a given spin-orbital index into its corresponding spatial atom label.

Parameters:

i (int) – The index of the spin-orbital.

Returns:

The atom label corresponding to the provided spin-orbital index.

This is in the form “U<num>” for upper atoms or “L<num>” for lower atoms, where <num> is the atom number.

Return type:

str

kappa(i: str, j: str)

Computes the screening parameter based on the parameter type.

Parameters:

• i (str) – Label for the first atom.

• j (str) – Label for the second atom.

Returns:

The screening parameter value.

Return type:

float

Raises:

ValueError – If the parameter type is invalid.

property n_electrons: int

property n_qubits: int

property num_spin_orbitals: int

Returns: int: The total number of spin-orbitals in the polyacene molecule.

ref_state()

Constructs the reference (singlet) state as a superposition of Hartree-Fock states with even and odd parities.

Returns:

The reference state as a dictionary of BinaryFockVector

to their respective coefficients.

Return type:

Dict[BinaryFockVector, float]

spin_idx(atom: str, spin: bool | int)

Calculates the index of the spin orbital associated with a given atom and spin.

Parameters:

• atom (str) – The label of the atom, where “U” represents an upper atom and “L” represents a lower atom, followed by a numerical index. Example: “U0”, “L1”.

• spin (Union[bool, int]) – The spin designation for the orbital. Acceptable values are either: - 0 or False for SPIN_DOWN - 1 or True for SPIN_UP

Returns:

The index of the spin orbital corresponding to the specified atom

and spin.

Return type:

int

Raises:

ValueError – If the atom label does not start with “L” or “U”.

u()

Computes the on-site repulsion term based on the parameter type.

Returns:

The on-site electron-electron repulsion in Hartree.

Return type:

float

Raises:

ValueError – If the parameter type is invalid.

v(i: str, j: str)

Computes the Coulomb interaction between two atomic sites.

Parameters:

• i (str) – Label for the first atom.

• j (str) – Label for the second atom.

Returns:

The Coulomb interaction value in Hartree, accounting for screening and distance-dependent effects between the atoms.

Return type:

float